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Dispersive limit from the Kawahara to the KdV equation

Abstract : We investigate the limit behavior of the solutions to the Kawahara equation $$ u_t +u_{3x} +\varepsilon u_{5x} + u u_x =0 , $$ as $ 0<\varepsilon \to 0 $. In this equation, the terms $ u_{3x} $ and $ \varepsilon u_{5x} $ do compete together and do cancel each other at frequencies of order $ 1/\sqrt{\varepsilon} $. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $ C([0,T];H^1(\R)) $ towards the solutions of the KdV equation for any fixed $ T>0$.
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https://hal.archives-ouvertes.fr/hal-00694082
Contributor : Luc Molinet <>
Submitted on : Thursday, June 7, 2012 - 11:27:55 AM
Last modification on : Thursday, March 5, 2020 - 5:33:29 PM
Document(s) archivé(s) le : Saturday, September 8, 2012 - 6:05:07 AM

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  • HAL Id : hal-00694082, version 2
  • ARXIV : 1205.0729

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Luc Molinet, Yuzhao Wang. Dispersive limit from the Kawahara to the KdV equation. Journal of Differential Equations, Elsevier, 2013, 255 (8), pp.2196-2219. ⟨hal-00694082v2⟩

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